I'm studying general topology and, more exactly, an introduction to cardinal functions. I'm following the Chapter I, called Cardinal Functions I, of the Handbook of Set-Theoretic Topology by Kenneth Kunen and Jerry E. Vaughan . The part where I'm stuck is in the proof of the Theorem 4.7:
4.7. Theorem (Hajnal-Juhász). For $X$, a topological Hausdorff space, $|X|\leq 2^{c(X)\chi(X)}$ where $c(X)$ is the cellularity of $X$ and $\chi(X)$ is the character of $X$.
The proof in the book is the next:
The hard part is the construction of the two sequences $\{A_\alpha : 0\leq \alpha<\kappa^+ \}$ and $\{\mathscr{V}_\alpha:0<\alpha<\kappa^+ \}$ because literally, they don't put any proof of how can we do the recursive construction. Assuming that construction, then the proof is easy and I don't have any problem. But, how can I do the recursive construction? Reading the book, I found the next proof that contains a "similar" recursive construction, but differs a lot from the proof that I need.
By the way, the 4.4 cited in the proof above is the next.
4.4. Theorem (Pospíŝil). If $X$ is a $T_2$ space then $|X|\leq d(X)^{\chi(X)}$
How can I do it the recursive construction? Any hint? I really apreciate any help you can provide me. Thanks a lot!
There is not much difference: going from $\alpha$ to $\alpha+1$ pick, for every subfamily $\mathcal{G}$ of $\mathcal{V}_\alpha$ that is of cardinality at most $\kappa$ and satifies $X\neq\overline{\bigcup\mathcal{G}}$ one point $x_\mathcal{G}$ in the complement $X\setminus\overline{\bigcup\mathcal{G}}$. This adds at most $2^\kappa$ many points to $A_\alpha$ and that's it; no need to show that the result is closed or take the closure of it.